I am trying to understand a certain proof of Catalan numbers and I do not understand a math behind one part

Question

The proof got that $$ \frac{1}{k+1}\frac{(2k)!}{(k!)^2}=\frac{1}{k+1}{2k \choose k} $$ Which indeed is the formula for Catalan numbers but I do not understand how: $$ \frac{(2k!)}{(k!)^2}={2k \choose k} $$

Thanks for help in advance.

Answer 1

You can apply the definition of binomial coefficients $\binom{n}{\ell}$ with $n=2k$ and $\ell=k$:

\begin{equation*} \binom{n}{\ell}=\frac{n!}{\ell!(n-\ell)!}=\frac{(2k)!}{k!(2k-k)!}=\frac{(2k)!}{k!k!}. \end{equation*}

Best

Answer 2

It follows because

$${2k\choose k}=\dfrac{2k!}{k! (2k-k)!}=\dfrac{2k!}{k!\times k!}. $$